In this post, I attempt to present two major metaphysical accounts of space by Kant and Leibniz, then present some recent findings from cognitive neuroscience about the neural basis of spatial cognition in an attempt to understand more about the nature of space and the possible connection of philosophical theories to empirical observations.

Immanuel Kant’s account of space in his Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based on the form of our a priori intuition (§9). As such, we are only able to intuit and perceive things in the world within the framework naturally provided by the capabilities and character (literally the under–standing) of our understanding. Kant then takes our intuitions of space (and time) as concepts integral to pure mathematics and as necessary components of our intuition (§10.91).

Kant develops that geometry is based on this pure intuition of space (and arithmetic on that of time) and advances that even after removing all sensations and empirical intuitions, the intuitions of space and time remain, proving them as the a priori intuitions that precede any form of empirical experience or sensation (ibid.103). Thus, our experience of space and the means by which we do geometry is a component of our intuition for Kant and does not require the existence of direct objects of experience. Rather, our awareness of things as they appear in space is woven into our intuition and is a basic characteristic of our experience. Kant goes on to describe space as the “form of outer intuition of our sensibility” in that it is the thing in which we perceive things, i.e. that it is a transcendental condition for sensation (§13.317). By this, we arrive at our understanding of the arrangements of objects in the world not by an empirical encounter, but by the form of our intuition. Therefore, Kant’s account makes geometry an intuitive practice that utilizes a basic component of our pure a priori intuition as opposed to our rational activity. In support, Kant offers that we determine a geometrical concept, e.g. congruency, not through concepts formed by reason, but through relations that are apparent as a result of our pure intuition (ibid.325).

Kant’s theory stands in stark contrast to that of Leibniz, whose account of space is intelligible through arguments in his Discourse on Metaphysics and Monadology. In the former Leibniz foreshadows the concept of the monad in arguing, “each singular substance expresses the universe in its own way,” which develops that the constituent, most fundamental components of reality itself are unique and infinite in number and contain all things present, past and possible (DM.9). In the latter work, Leibniz reiterates that each monad must be different from each other because no two monads can be identical, further establishing the notion of an infinite number of infinite substances that make up the universe (M8-9). Accordingly, objects in the world are made up of monads, which are self contained and distinct from one another. Based on Leibniz’s theory, we arrive at a higher order reality in which everything is separate, distinct and self-contained and therefore, space comes about as a consequence of the existence of objects. That is, when we perceive Leibnizian space, we perceive a thing produced as a result of the existence of two other separate objects.

Leibniz’s account of space also has implications for geometry. By his theory, our perceptions of congruent things for example, become a comparison of two objects made in perception and understood actively by reason. Evidence for this can be found in Leibniz’s arguments concerning physics and causality. Leibniz believes that God constructed the world (with monads) and everything in it in the best possible way (M1,3). As such, the universe carries a predetermined and pre-established order of cause and effect (M6,7) and Leibniz argues that we come to understand nature by finding causes from effects by the use of reason (DM19). Therefore, geometry becomes a rational activity when viewed from a Leibnizian perspective because it is an investigation of that which exists. For Leibniz, we must necessarily invoke our understanding of the nature of objects in the world to do geometry, and this conflicts with the intuitive nature that Kant ascribes to geometry. For Kant, our knowledge (or ‘cognition’) of space is a result of the form of our intuition that comes before sensibility, which makes our understanding of geometry intuitive. For Leibniz, space exists only because discrete objects exist in the world and our understanding of geometry comes from rational manipulations of those objects. Nevertheless, both Kant and Leibniz provide accounts for space that necessarily involve an a priori component rather than perception alone.

Cognitive neuroscientists are now suggesting that spatial cognition is a complex interaction of multiple brain circuits in parallel that make use of both allocentric and egocentric processing of the external world. A pivotally important concept in understanding spatial cognition has been the investigation of representation in the brain. “Representation” is a term that has been used in philosophy for centuries, and science is now using the term to refer to the neural picture of the external world as observed by our brain monitoring and imaging technology. The investigation of representation in the brain essentially involves solving the puzzle of how the world itself is represented physically in the brain. With respect to spatial cognition, the discovery of grid cells in 2005 suggests that a euclidean space is encoded in the brain itself by neurons, and that activation and deactivation of grid cells plays a major role in representing the spatiality of the external world to the perceiver. The discovery of grid cells also suggested a mechanism for the perception of one’s own location that is continually updated by input from the external world, suggesting that, similar to visual perception, the representation of space in the brain itself is an active phenomenon that varies just as much as the visual field.

Most, if not all, of the work on spatial perception and grid cells is performed on rats and conclusions made are inductively applied to humans, which makes us doubt how accurately these mechanisms can apply to the human brain. I say this because while many other studies of physiological phenomena in rats or mice (e.g. those on the cardiovascular and immune systems) may be stronger due to the increased homology to humans present in those systems. In other words, I think that the human and the mouse/rat brain differ quite significantly, perhaps more so than other organs and that this reduces the strength of our inductive conclusions. However, interesting studies are now being performed on humans which place a subject in a virtual maze through a computer program and measure brain activity through noninvasive methods such as functional MRI (fMRI). Many recent studies are pointing to the hippocampus as a major player in way finding and general navigation through virtual mazes, which suggests that our spatial perception is an evolutionarily refined phenomenon, but also one that is fundamental to our basic neural make up. Interestingly, the neural phenomena change when scientists investigate spatial cognition relative to landmarks (i.e. objects) as compared to studies in simple maze navigation. In these object-centric experiments, subjects navigated mazes and were cued with objects present in the virtual environment that they had to collect and place in a distinct virtual location, either at a specific landmark or in a general bounded area. Brain scans in these studies showed both hippocampal and striatal activation during the performed tasks, with hippocampal activity associated with the boundary task and striatal activity associated with the landmark task. Further, separate studies in rats performing similar spatial boundary tasks reveal that the activation of hippocampal “place cells” fire in boundary-space tasks, which scientists think are creating a matched representation of distances and angles relative to the boundaries in the visual field. Results from striatal activation are still unclear and are being more closely investigated. It has also been suggested that the hippocampal and striatal circuits act in parallel rather than in series or in combination. This makes sense given that spatial cognition may involve both boundary and landmark elements, as when we have to hammer a nail into a specific location or plug something into a power outlet.

Relating the philosophy and neuroscience presented in this post, it seems that both Kantian and Leibnizian conceptions of space are compatible with neuroscientific findings about spatial cognition. Kant’s theory applies to the current understanding of hippocampal, boundary influenced tasks in that both suggest a holistic conception of space – that is, space can be understood as object independent. On the other hand, Leibnizian conceptions of space and the landmark results suggest a more object-dependent framework for spatial cognition. As spatial cognition and perception are our most direct means to accessing and interacting with the external world, both scientists and philosophers of the future ought to work together on this enormously complex problem in an effort to postulate how spatial phenomena as presented to us by the mind relate to neural phenomena in the brain. Perhaps then we will move closer to filling the explanatory gap between the mind and brain.